#133 The highway between Atlanta, Georgia, and Athens

The highway between Atlanta, Georgia, and Athens - Math

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Free Chegg Question

he highway between Atlanta, Georgia and Athens, Georgia has a high incidence of accidents along its 100 kilometers. Public safety officers say that the occurrence of accidents along the highway is randomly (uniformly) distributed, but the news media say otherwise. The Georgia Department of Public Safety published records for the month of September These records indicated the point at which 30 accidents involving an injury or death occurred, as follows (the data points representing the distance from the city limits of Atlanta):

Use the Kolmogorov-Smirnov test to discover whether the distribution of location of accidents is uniformly distributed for the month of September.
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Free Chegg Answer

The pdf of uniform distribution is f(x) = 1 / (30-1) = 1/29   for 1 <= x <= 30
H0: F(X)=F₀(x)
H1: F(X) not=F₀(x)
where F(x) is the (unknown) cdf from which our data were sampled, and
The CDF of uniform distribution is F₀(X) = x/29              1<x<30
Now, in working towards calculating dn, we first need to order the eight data points so that y1≤⋯≤y30. The table below provides all the necessary values for finding the KS test statistic. Note that the empirical cdf satisfies Fn(yk)=k/30, for k=0,1,…,30

k	yk	Fn(yk-1)	Fn(yk)	F0(yk))	|Fn(yk-1)-F0(yk))|	|Fn(yk)-F0(yk))|
1	6	0	0.033333	0.206897	0.206896552	0.173563218
2	7	0.033333	0.066667	0.241379	0.208045977	0.174712644
3	17.2	0.066667	0.1	0.593103	0.526436782	0.493103448
4	20.6	0.1	0.133333	0.710345	0.610344828	0.577011494
5	21.6	0.133333	0.166667	0.744828	0.611494253	0.57816092
6	23.3	0.166667	0.2	0.803448	0.636781609	0.603448276
7	23.7	0.2	0.233333	0.817241	0.617241379	0.583908046
8	27.3	0.233333	0.266667	0.941379	0.708045977	0.674712644
9	27.3	0.266667	0.3	0.941379	0.674712644	0.64137931
10	32.4	0.3	0.333333	1.117241	0.817241379	0.783908046
11	36.3	0.333333	0.366667	1.251724	0.918390805	0.885057471
12	36.8	0.366667	0.4	1.268966	0.902298851	0.868965517
13	40.7	0.4	0.433333	1.403448	1.003448276	0.970114943
14	45.2	0.433333	0.466667	1.558621	1.125287356	1.091954023
15	45.3	0.466667	0.5	1.562069	1.095402299	1.062068966
16	62.6	0.5	0.533333	2.158621	1.65862069	1.625287356
17	67.3	0.533333	0.566667	2.32069	1.787356322	1.754022989
18	69.8	0.566667	0.6	2.406897	1.840229885	1.806896552
19	73.1	0.6	0.633333	2.52069	1.920689655	1.887356322
20	73.2	0.633333	0.666667	2.524138	1.890804598	1.857471264
21	76.6	0.666667	0.7	2.641379	1.974712644	1.94137931
22	87.2	0.7	0.733333	3.006897	2.306896552	2.273563218
23	87.6	0.733333	0.766667	3.02069	2.287356322	2.254022989
24	87.8	0.766667	0.8	3.027586	2.26091954	2.227586207
25	88.3	0.8	0.833333	3.044828	2.244827586	2.211494253
26	90.1	0.833333	0.866667	3.106897	2.273563218	2.240229885
27	91.7	0.866667	0.9	3.162069	2.295402299	2.262068966
28	97.4	0.9	0.933333	3.358621	2.45862069	2.425287356
29	98.8	0.933333	0.966667	3.406897	2.473563218	2.440229885
30	99.7	0.966667	1	3.437931	2.471264368	2.437931034

The last two columns represent all the differences we need to check. The largest of these is d30=2.47356. From KS Table with α=0.05, the critical value is 0.24. So, we reject the claim that the data was not sampled from Uniform(1,30).
i.e. The distribution of location of accidents is not uniformly distributed for the month of september